| L(s) = 1 | + (0.844 + 1.13i)2-s + (0.0447 + 1.73i)3-s + (−0.572 + 1.91i)4-s + (−4.00 − 1.45i)5-s + (−1.92 + 1.51i)6-s + (1.46 + 0.258i)7-s + (−2.65 + 0.970i)8-s + (−2.99 + 0.154i)9-s + (−1.73 − 5.77i)10-s + (1.57 + 4.33i)11-s + (−3.34 − 0.904i)12-s + (2.82 + 3.36i)13-s + (0.946 + 1.88i)14-s + (2.34 − 6.99i)15-s + (−3.34 − 2.19i)16-s + (2.43 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (0.597 + 0.801i)2-s + (0.0258 + 0.999i)3-s + (−0.286 + 0.958i)4-s + (−1.79 − 0.651i)5-s + (−0.786 + 0.617i)6-s + (0.554 + 0.0977i)7-s + (−0.939 + 0.343i)8-s + (−0.998 + 0.0516i)9-s + (−0.547 − 1.82i)10-s + (0.475 + 1.30i)11-s + (−0.965 − 0.261i)12-s + (0.784 + 0.934i)13-s + (0.252 + 0.503i)14-s + (0.605 − 1.80i)15-s + (−0.836 − 0.548i)16-s + (0.590 − 0.340i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.189585 + 1.14346i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.189585 + 1.14346i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.844 - 1.13i)T \) |
| 3 | \( 1 + (-0.0447 - 1.73i)T \) |
| good | 5 | \( 1 + (4.00 + 1.45i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 0.258i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-1.57 - 4.33i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.82 - 3.36i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.43 + 1.40i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.516 + 0.895i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.345 + 1.95i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.783 - 0.657i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.04 + 1.06i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (5.15 - 2.97i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.77 - 3.30i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.28 - 0.830i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.806 - 4.57i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 + (-2.77 + 7.62i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.25 - 0.397i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.24 + 6.07i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.21 - 3.84i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.12 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.19 - 2.62i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.23 + 3.85i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (3.14 + 1.81i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.20 - 1.89i)T + (74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.55057771595595079009968056633, −11.80914416781310026892562431576, −11.26673582641376247236402707756, −9.519078866990752654318491665463, −8.545546524427752987451365578234, −7.88266885768990220302035127591, −6.68734915999053380836363943679, −4.89191651681760133199093418894, −4.46567145249808348344115860092, −3.50336953665706652549197155883,
0.885269461971376804774059412786, 3.08912319867720464628466604749, 3.81753252429590508986316184674, 5.57352447189955704148705538845, 6.72020607229496412897304485025, 8.010165212690857701385006935428, 8.535581821219461127319990700882, 10.57151549908966667876110651495, 11.26688258529036179055753844085, 11.80161141712521112719538737020
